A unifying functional approach for inconsistency indices of pairwise comparisons

A vast amount of indices has been proposed in the literature [1] to quantify the deviation of a pairwise comparison matrix A = (aij)n×n from the consistency property aik = aijajk ∀i, j, k. In this paper we propose a unifying approach by expressing an inconsistency index as a function of the matrix entries aij satisfying a set of suitable properties. More precisely, we first consider the ‘local’ inconsistency concerning three alternatives Ai, Aj , Ak and we define its general form by means of a function f(aij , ajk, aki) of the three preference intensities aij , ajk, aki. Then, we aggregate the contributions of local inconsistencies by means of a suitable aggregation function ⊕ in order to obtain a global inconsistency index ⊕n i<j<k f(aij , ajk, aki) for matrix A. It is interesting to note that almost all the known inconsistency indices are particular cases of the general form described above. We cite, as examples, the Geometric Consistency Index [4] and the indices introduced by (a) Peláez and Lamata [7], (b) Cavallo and D’Apuzzo [3], (c) Koczkodaj [5, 6]. We study which properties should satisfy functions f and ⊕ in order to generate satisfactory inconsistency indices. In particular, we determine a set of properties for f and ⊕ which guarantees that the obtained index satisfies six axiomatic properties recently introduced in order to characterize an inconsistency index [2].